Solution of the time-harmonic viscoelastic inverse problem with interior data in two dimensions

We consider the problem of determining the distribution of the complex-valued shear modulus for an incompressible linear viscoelastic material undergoing infinitesimal time-harmonic deformation, given the knowledge of the displacement field in its interior. In particular, we focus on the two-dimensional problems of anti-plane shear and plane stress. These problems are motivated by applications in biomechanical imaging, where the material modulus distributions are used to detect and/or diagnose cancerous tumors. We analyze the well-posedness of the strong form of these problems and conclude that for the solution to exist, the measured displacement field is required to satisfy rather restrictive compatibility conditions. We propose a weak, or a variational formulation, and prove the existence and uniqueness of solutions under milder conditions on measured data. This formulation is derived by weighting the original PDE for the shear modulus by the adjoint operator acting on the complex-conjugate of the weighting functions. For this reason, we refer to it as the complex adjoint weighted equation (CAWE). We consider a straightforward finite element discretization of these equations with total variation regularization, and test its performance with synthetically generated and experimentally measured data. We find that the CAWE method is, in general, less diffusive than a corresponding least squares solution, and that the total variation regularization significantly improves its performance in the presence of noise.